On Hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$

TL;DR

This paper classifies various special hypersurfaces in the product space , including those with constant principal curvatures, homogeneous, isoparametric, and with limited principal curvatures, advancing understanding of their geometric properties.

Contribution

It provides a comprehensive classification of hypersurfaces in  with specific curvature and symmetry conditions, including new results on those with limited principal curvatures.

Findings

Classified hypersurfaces with constant principal curvatures

Identified homogeneous and isoparametric hypersurfaces

Characterized hypersurfaces with up to three constant principal curvatures

Abstract

In this paper, we study hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$. We first classify the hypersurfaces with constant principal curvatures and constant product angle function. Then, we classify homogeneous hypersurfaces and isoparametric hypersurfaces, respectively. Finally, we classify the hypersurfaces with at most two distinct constant principal curvatures, as well as those with three constant principal curvatures under some additional conditions.